Following the standard Rudin-Osher–Fatemi model in [1]}, the operator \(\text{Prox}_{\frac{\alpha }{\beta } R(x)}\)  can be interpreted as the denoising operator for the standard image denoising model. In the last few years, the literature of algorithms for various image denoising models has been phenomenally upgraded by contemporary deep neural networks, see, e.g.,[2]}, [3]}. An advantage of applying a deep neural network to denoising models is that it avoids iterations in its testing phase,
and hence computation can be largely saved. We are thus inspired to consider some pre-trained deep neural network, rather than some iterative scheme, for the \(p_h\) -subproblem (REF ).
